Math / Calculus

Question7. The largest interval in which a solution of the IVP $y^{\prime}+(\ln t) y=\tan t, \quad y\left(\frac{\pi}{4}\right)=1$ is certain to exist is (a) $\left(0, \frac{\pi}{2}\right)$ (b) $(0, \pi)$ (c) $(0,1)$ . (d) $(1, \infty)$

Studdy Solution

STEP 1

1. The given initial value problem (IVP) is a first-order linear differential equation.
2. The existence and uniqueness of solutions for first-order linear differential equations depend on the continuity of the coefficients and the initial condition.
3. We need to identify the interval where the coefficients of the differential equation are continuous.

STEP 2

1. Identify the coefficients of the differential equation.
2. Determine the intervals of continuity for each coefficient.
3. Find the largest interval where all coefficients are continuous and include the initial condition.
4. Compare the identified interval with the given options.

STEP 3

Identify the coefficients of the differential equation:
The differential equation is given by:
$y^{\prime} + (\ln t) y = \tan t$
The coefficients are $\ln t$ and $\tan t$ .

STEP 4

Determine the intervals of continuity for each coefficient:
1. The function $\ln t$ is continuous for $t > 0$ .
2. The function $\tan t$ is continuous for $t \neq \frac{\pi}{2} + n\pi$ , where $n$ is an integer.

STEP 5

Find the largest interval where all coefficients are continuous and include the initial condition:
The initial condition is given at $t = \frac{\pi}{4}$ .
1. $\ln t$ is continuous for $t > 0$ .
2. $\tan t$ is continuous for $t \neq \frac{\pi}{2}$ .
The largest interval that includes $t = \frac{\pi}{4}$ and satisfies both conditions is $\left(0, \frac{\pi}{2}\right)$ .

STEP 6

Compare the identified interval with the given options:
The identified interval $\left(0, \frac{\pi}{2}\right)$ matches option (a).
The largest interval in which a solution of the IVP is certain to exist is $\boxed{\left(0, \frac{\pi}{2}\right)}$ .

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Question7. The largest interval in which a solution of the IVP $y^{\prime}+(\ln t) y=\tan t, \quad y\left(\frac{\pi}{4}\right)=1$ is certain to exist is (a) $\left(0, \frac{\pi}{2}\right)$ (b) $(0, \pi)$ (c) $(0,1)$ . (d) $(1, \infty)$

Studdy Solution

STEP 1

STEP 2

1. Identify the coefficients of the differential equation.
2. Determine the intervals of continuity for each coefficient.
3. Find the largest interval where all coefficients are continuous and include the initial condition.
4. Compare the identified interval with the given options.

STEP 3

Identify the coefficients of the differential equation:
The differential equation is given by:
$y^{\prime} + (\ln t) y = \tan t$
The coefficients are $\ln t$ and $\tan t$ .

STEP 4

Determine the intervals of continuity for each coefficient:
1. The function $\ln t$ is continuous for $t > 0$ .
2. The function $\tan t$ is continuous for $t \neq \frac{\pi}{2} + n\pi$ , where $n$ is an integer.

STEP 5

STEP 6

Compare the identified interval with the given options:
The identified interval $\left(0, \frac{\pi}{2}\right)$ matches option (a).
The largest interval in which a solution of the IVP is certain to exist is $\boxed{\left(0, \frac{\pi}{2}\right)}$ .

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Studdy Solution

STEP 1

STEP 2

1. Identify the coefficients of the differential equation.2. Determine the intervals of continuity for each coefficient.3. Find the largest interval where all coefficients are continuous and include the initial condition.4. Compare the identified interval with the given options.

STEP 3

Identify the coefficients of the differential equation:The differential equation is given by:y′+(ln⁡t)y=tan⁡t y^{\prime} + (\ln t) y = \tan t y′+(lnt)y=tantThe coefficients are ln⁡t \ln t lnt and tan⁡t \tan t tant.

STEP 4

Determine the intervals of continuity for each coefficient:1. The function ln⁡t \ln t lnt is continuous for t>0 t > 0 t>0.2. The function tan⁡t \tan t tant is continuous for t≠π2+nπ t \neq \frac{\pi}{2} + n\pi t=2π​+nπ, where n n n is an integer.

STEP 5

STEP 6

Compare the identified interval with the given options:The identified interval (0,π2) \left(0, \frac{\pi}{2}\right) (0,2π​) matches option (a). The largest interval in which a solution of the IVP is certain to exist is (0,π2) \boxed{\left(0, \frac{\pi}{2}\right)} (0,2π​)​.

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1. Identify the coefficients of the differential equation.
2. Determine the intervals of continuity for each coefficient.
3. Find the largest interval where all coefficients are continuous and include the initial condition.
4. Compare the identified interval with the given options.

Identify the coefficients of the differential equation:
The differential equation is given by:
$y^{\prime} + (\ln t) y = \tan t$
The coefficients are $\ln t$ and $\tan t$ .

Determine the intervals of continuity for each coefficient:
1. The function $\ln t$ is continuous for $t > 0$ .
2. The function $\tan t$ is continuous for $t \neq \frac{\pi}{2} + n\pi$ , where $n$ is an integer.

Compare the identified interval with the given options:
The identified interval $\left(0, \frac{\pi}{2}\right)$ matches option (a).
The largest interval in which a solution of the IVP is certain to exist is $\boxed{\left(0, \frac{\pi}{2}\right)}$ .